We study the existence of special points in extremally disconnected compact topological spaces that witness their nonhomogeneity. Via Stone duality, we are looking for ultrafilters on complete Boolean algebras with special combinatorial properties. We introduce the notion of a coherent ultrafilter (coherent P-point, coherently selective). We show that generic existence of such ultrafilters on every complete ccc Boolean algebra of weight not exceeding the continuum is consistent with set theory, and that they witness the nonhomogeneity of the corresponding Stone spaces. We study the properties of the order-sequential property on σ-complete Boolean algebras and its relation to measure-theoretic properties. We ask whether the order-sequential topology can be compact in a nontrivial case, and partially answer the question in a special case of the Suslin algebra associated with a Suslin tree.

Orientador: Prof. Dr. Eduardo Guéron / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional - PROFMAT, Santo André, 2018. / Calculadoras são aparelhos comuns no cotidiano do homem moderno, contudo, os conceitos matemáticos envolvidos em sua concepção ainda são conhecidos por poucos. Durante séculos, a obstinação da humanidade em construir máquinas capazes de computar de forma autônoma resultou tanto no surgimento dos atuais computadores, como também em um magnífico legado de conhecimentos matemáticos agregados a tal conquista. Conteúdos tais como congruências e álgebra booleana suscitaram a revolução dos sistemas informatizados e tem sido amplamente explorados por meio de inúmeras aplicações, nossa trajetória perpassou pela aritmética modular, o teorema de Euler-Fermat e as classes residuais, além de bases numéricas, tópicos de eletrônica digital e funções booleanas, com foco no desenvolvimento de circuitos lógicos e o engendrar de componentes eletrônicos, que configuram a base para idealização e construção de calculadoras que efetuem as operações aritméticas em bases arbitrárias, objetivo preponderante deste trabalho. O esmiuçar das etapas de construção das calculadoras, viabiliza o aprofundamento dos conceitos matemáticos que a fomentaram. A abordagem dos temas supracitados culmina para aprimorar e evidenciar a aplicabilidade da matemática à essência da era moderna. / Calculators are common apparatuses in the everyday of modern man, however, the mathematical concepts involved in its conception are still known by few. For centuries, mankind¿s obstinacy in building machines capable of computing autonomously resulted in both the emergence of current computers and a magnificent legacy of mathematical knowledge added to such achievement. Contents such as congruences and Boolean algebra have aroused the revolution of computerized systems and it has been extensively explored through numerous applications, our trajectory ran through modular arithmetic, Euler-Fermat¿s theorem and residual classes, as well as numerical bases, topics of digital electronics and Boolean functions, focusing on the development of logic circuits and the generation of electronic components, which form the basis for the design and construction of calculators that perform arithmetic operations on arbitrary bases, a preponderant objective of this work. The to detail of the construction steps of the calculators, enables the deepening of the mathematical concepts that fomented it. The approach to the aforementioned themes culminates in improving and evidencing the applicability of mathematics to the essence of the modern era.

Simple counting quantifiers that can be used to compare the number of role successors of an individual or the cardinality of a concept with a fixed natural number have been employed in Description Logics (DLs) for more than two decades under the respective names of number restrictions and cardinality restrictions on concepts. Recently, we have considerably extended the expressivity of such quantifiers by allowing to impose set and cardinality constraints formulated in the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA) on sets of role successors and concepts, respectively. We were able to prove that this extension does not increase the complexity of reasoning. In the present paper, we investigate the expressive power of the DLs obtained in this way, using appropriate bisimulation characterizations and 0–1 laws as tools to differentiate between the expressiveness of different logics. In particular, we show that, in contrast to most classical DLs, these logics are no longer expressible in first-order predicate logic (FOL), and we characterize their first-order fragments. In most of our previous work on DLs with QFBAPA-based set and cardinality constraints we have employed finiteness restrictions on interpretations to ensure that the obtained sets are finite, as required by the standard semantics for QFBAPA. Here we dispense with these restrictions to ease the comparison with classical DLs, where one usually considers arbitrary models rather than finite ones, easier. It turns out that doing so does not change the complexity of reasoning.

We introduce a new description logic that extends the well-known logic ALCQ by allowing the statement of constraints on role successors that are more general than the qualified number restrictions of ALCQ. To formulate these constraints, we use the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA), in which one can express Boolean combinations of set constraints and numerical constraints on the cardinalities of sets. Though our new logic is considerably more expressive than ALCQ, we are able to show that the complexity of reasoning in it is the same as in ALCQ, both without and with TBoxes. / The first version of this report was put online on April 6, 2017. The current version, containing more information on related work, was put online on July 13, 2017. This is an extended version of a paper published in the proceedings of FroCoS 2017.